Open Archive TOULOUSE Archive Ouverte (OATAO) OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 18454 To link to this article : DOI:10.1109/20.123957 URL : http://dx.doi.org/10.1109/20.123957 To cite this version : Sadowski, Nelson and Lefevre, Yvan and Lajoie- Mazenc, Michel and Cros, Jerome Finite element torque calculation in electrical machines while considering the movement. (1992) IEEE Transactions on Magnetics, vol. 28 (n° 2). pp. 1410-1413. ISSN 0018- 9464 Any correspondence concerning this service should be sent to the repository administrator: [email protected] FINITE ELEMENT TORQUE CALCULATION IN ELECTRICAL MACHINES WHILE CONSIDERING THE MOUVEMENT N.Sadowski(*) Y.Lefèvre M.Lajoie-Mazenc J.Cros LEEI/ENSEEIHT 2, Rue Camichel 31.071 Toulouse Cedex FRANCE (*)DEEL/CTC/UFSC BRAZIL - Currently with the LEEI/Toulouse/France Al)stract: In this paper different methods are layer can occupy all the airgap region or presented for the calculation of torque as a just a part of it. This technique is known as function of rotation angle in an electrical the Moving band method [3]. At each step of machine. These methods are integrated in a. the moving part displacement, the elements of calculation code by using the finite element the Moving band are distorted and when the method. The movement is taken into account by distorsion is large enough, then the airgap means of the technique of the Moving Band, is remeshed. In this way, the nurnber of constituted by quadrilateral finite elements demain nodes can increase, but by using a in the airgap. The torque is calculated dynamic allocation of the periodicity or during the displacement of the moving part by anti-periodicity conditions, the size of the using the following methods: Maxwell stress corresponding matrices is not increased. tensor, Coenergie derivation, Coulomb's v'irtual work, Arkkio' s method and the As refered to the computation time, the Magnetizing current method. The results Moving band method is much faster than the obtained by the different methods are methods in which the airgap is net compared with experimental data and.allow te discretized. For this reason we have chosen deduce practical informat,ions concerning the this method for our study. By considering the advantages and limitations of each method. importance of an adequate representation of the airgap in which the Moving band is INTRODUCTION placed, we have chosen quadrilateral elements for its composition (fig.1). Actually the design of electrical machines involves increasingly the While the rotor displacement is equal to calculation codes of the electromagnetic the Moving band discretization step, the fields based on the fini te element method. integration of these quadrilateral elements The knowledge of the torque variations in is carried out by dividing them into four terms of the rotation angle of an electrical triangles, as illustrated in figure 2. machine is very important for the designer. However, if the displacement is equal to a In fact, this knowledge allows him to fraction of the discretization step, this evaluate the performances and operating division in the two directions can yield a qualities of the machine which is te be pair of triangles more flattened than the designed. The determination of the torque other pair, as shown in figure 3. In this variations as a function of position, case, between all the elements which compose involves two coupled techniques: the Moving band, we look for the triangles that can be derived from the quadrilateral a) Movement included methods. element, and for the integration we will b) The torque calculation method which is the retain the division into two triangles with best adapted to the movement included method. the largest quality factor [4]. This paper studies this coupled problem, examines the different techniques of torque calculation and provides comparison elements for choosing the best method. band Considering the movement during the field calculations The techniques ta take into account the movement can be separated in two categories: 1) The airgap is not discretized; in this case, the airgap can be modeled by the boundary-element method [l] or by using analytical solutions [2]. Figure 1: Airgap and Moving band. 2) The airgap is discretized; here, the airgap is subdivided into meshes and the rotation can take place by means of a layer Methods of Torque calculation of finite elements placed in the airgap. This several methods have been presented in the literature for torque calculation: where Lis the length and 6 a represents the displacement. 4) Coulomb's Virtual Work: based on the Figure 2 : Division of quadrilateral element principle of virtual works [7], gives the when the displacement step is equal to the following expression for the torque: discretization step. t T= f L[-B G-l(dG/dt)H + Jn (5) + f�aHIGl-11dlGl/cttJJdn J 0 Figure 3: Division of quadrilateral element where the integration is carried out over the when the displacement step is different from elements situated between the fixed and the discretization step. moving parts, having undergone a virtual deformation. In equation (5), L is the 1) Maxwell stress tensor: the force applied length, G denotes the jacobian matrix, dG/dt to one part of the magnetic circuit can be is its derivative representing the element obtained by integrating the well-known deformation during the displacement d�, 1 G 1 Maxwell stress tensor along a surface r, is the determinant of Gand (d!Gl/dt) is the placed in the air and enclosing this part. In derivative of the determinant , representing the case of electrical machines, this surface the variation of the element volume during is normally placed in the airgap and the the displacement dt. torque can be calculated by the following relationship: 5) Maqnetizinq Current Method: this method is based on the calculation of the magnetizing current and the flux density over the element 2 T=Lf(r x [(1/µ0)(B,n)B -(l/2µ0JB nJJdr (1) edges which constitute the boundary between Jr the iron or permanent magnet and the air [8]. Here the torque can be determined by the where L is the length, B is the induction following expression: vector in the elements and r is the lever arm, i.e. the vector which connects the 2 2 origin ta the midpoint of segment ar. In T=(L/µ0) f1rx[(Bt1 -Bt2 )n order ta have a better accuracy, dr is drawn Jrc t t (6) by connecting the midpoints of he riangular 2 element edges [5]. When the quadrilateral -(Bt1Bt2-Bn2 )t]}drc elements are considered, the average torque is obtained by dividing the elements where Lis the machine length, rc denotes all according the two directions (fig.1). the interface between the iron or permanent magnet and the air, and drc is the length of 2 ) Arkkio's method: this is a variant of the the element edge located at the boundary. The Maxwell stress tensor and consists in vector ris the lever arm, that is the vector integrating the torque given by (1) in the which connects the origin ta the midpoint of whole volume of the airgap comprised between drc . Bti and Bni denote respectively, tangential and normal flux densities with the layers of radii rr and r5 • This method has been presented by Arkkio [ 6], with the respect to arc. The subscript 1 refers ta the following expression for the torque: iron or permanent magnet, while 2 corresponds to the air. (2) OBTAINED RESULTS AND DISCUSSION In order to compare the above mentioned t in which L is the length, Br and B� denote methods, two permanen magnet machine the radial and tangential inductions in the structures have been considered: elements of surface s, comprised between 1) The first structure (fig.4) is a motor radii rr and r5, and dS is the surface of one element. with three pole pairs and has, in the region under study, three blocs of permanent magnet, magnetized in parallel with the center magnet 3) Method of magnetic coenergy derivation: axis. For the purpose of numerical the torque can be calculated by deriving the calculations, the airgap mesh is composed of magnetic coenergy w, by maintaing the current two layers of quadrilateral elements, one of constant: which being the Moving band. The angular discretization step is taken equal to 1, 25 H degrees. T=LdW /da=d{f f B�tlll dVJ /da (3) We present two sets of simulation results JvJ o for the static torque obtained with two series phases, supplied by a constant DC In the numerical modeling, this derivation is current: when. the rotation step is equal to approximated by the difference between two the discretization step and when it is equal successive calculations, that is: ta on half of.the discretization step. a b Force IN) , ---- MT '�•c 1000 2000 3000 000 Number of Elements C Figure 4: First machine a) Rotation step egual to 1. 2 5 degrees: the Figure 6:a)Permanent inagnet and iron piece. corresponding simulation results are forces illustrated in figure 5. b)Surface distribution of jO obtained by the Magnetizing current method Torque (N.m) (MC) c)Global force, determined by the Maxwell stress tensor (MT) and Magnetizing current method (MC) in terms of number of 5 elements. b) Rotation step is egual to · o. 75 degrees: figure 7 shows the results obtained for this rotation step, which is the half of the Moving band discretization step.